The equation of hyperbola $H$ is $\dfrac {(y+3)^{2}}{25}-\dfrac {(x-4)^{2}}{4} = 1$. What are the asymptotes?
Solution: We want to rewrite the equation in terms of $y$ , so start off by moving the $y$ terms to one side: $\dfrac {(y+3)^{2}}{25} = 1 + \dfrac {(x-4)^{2}}{4}$ Multiply both sides of the equation by $25$ $(y+3)^{2} = { 25 + \dfrac{ (x-4)^{2} \cdot 25 }{4}}$ Take the square root of both sides. $\sqrt{(y+3)^{2}} = \pm \sqrt { 25 + \dfrac{ (x-4)^{2} \cdot 25 }{4}}$ $ y + 3 = \pm \sqrt { 25 + \dfrac{ (x-4)^{2} \cdot 25 }{4}}$ As $x$ approaches positive or negative infinity, the constant term in the square root matters less and less, so we can just ignore it. $y + 3 \approx \pm \sqrt {\dfrac{ (x-4)^{2} \cdot 25 }{4}}$ $y + 3 \approx \pm \left(\dfrac{5 \cdot (x - 4)}{2}\right)$ Subtract $3$ from both sides and rewrite as an equality in terms of $y$ to get the equation of the asymptotes: $y = \pm \dfrac{5}{2}(x - 4) -3$